Small Subgraphs Research Articles

Scale-free models for the structure of business firm networks

We study firm collaborations in the life sciences and the information and communication technology sectors. We propose an approach to characterize industrial leadership using k -shell decomposition, with top-ranking firms in terms of market value in higher k -shell layers. We find that the life sciences industry network consists of three distinct components: a "nucleus," which is a small well-connected subgraph, "tendrils," which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a "bulk body," which consists of the majority of nodes. Industrial leaders, i.e., the largest companies in terms of market value, are in the highest k -shells of both networks. The nucleus of the life sciences sector is very stable: once a firm enters the nucleus, it is likely to stay there for a long time. At the same time we do not observe the above three components in the information and communication technology sector. We also conduct a systematic study of these three components in random scale-free networks. Our results suggest that the sizes of the nucleus and the tendrils in scale-free networks decrease as the exponent of the power-law degree distribution lambda increases, and disappear for lambda>or=3 . We compare the k -shell structure of random scale-free model networks with two real-world business firm networks in the life sciences and in the information and communication technology sectors. We argue that the observed behavior of the k -shell structure in the two industries is consistent with the coexistence of both preferential and random agreements in the evolution of industrial networks.

SING: Subgraph search In Non-homogeneous Graphs

BackgroundFinding the subgraphs of a graph database that are isomorphic to a given query graph has practical applications in several fields, from cheminformatics to image understanding. Since subgraph isomorphism is a computationally hard problem, indexing techniques have been intensively exploited to speed up the process. Such systems filter out those graphs which cannot contain the query, and apply a subgraph isomorphism algorithm to each residual candidate graph. The applicability of such systems is limited to databases of small graphs, because their filtering power degrades on large graphs.ResultsIn this paper, SING (Subgraph search In Non-homogeneous Graphs), a novel indexing system able to cope with large graphs, is presented. The method uses the notion of feature, which can be a small subgraph, subtree or path. Each graph in the database is annotated with the set of all its features. The key point is to make use of feature locality information. This idea is used to both improve the filtering performance and speed up the subgraph isomorphism task.ConclusionsExtensive tests on chemical compounds, biological networks and synthetic graphs show that the proposed system outperforms the most popular systems in query time over databases of medium and large graphs. Other specific tests show that the proposed system is effective for single large graphs.

Counting Matchings and Tree-Like Walks in Regular Graphs

The number of closed tree-like walks in a graph is closely related to the moments of the roots of the matching polynomial for the graph. Thus, by counting these walks up to a given length it is possible to find approximations for the matching polynomial. This approach has been used in two separate problems involving asymptotic enumerations of 1-factorizations of regular graphs. Nevertheless, a systematic way to count the required walks had not previously been found.In this paper we give an algorithm to count closed tree-like walks in a regular graph up to a given length. For smallm, this provides expressions for the number ofm-matchings in the graph in terms of the numbers of copies of certain small subgraphs that appear in the graph. The simplest of these expressions were already known, having been rediscovered by numerous authors usingad hocmethods. We offer the first general method for producing the expressions. We also find generating functions that isolate the contribution from the simplest kind of subgraph – namely a single cycle of arbitrary length.

Variance of the subgraph count for sparse Erdős–Rényi graphs

We develop formulas for the variance of the number of copies of a small subgraph H in the Erdős–Rényi random graph. The central technique employs a graph overlay polynomial encoding subgraph symmetries, which is of independent interest, that counts the number of copies H ˜ ≅ H overlapping H . In the sparse case, building on previous results of Janson, Łuczak, and Rucinski allows restriction of the polynomial to the asymptotically contributing portion either when H is connected with non-null 2-core, or when H is a tree. In either case we give a compact computational formula for the asymptotic variance in terms of a rooted tree overlay polynomial. Two cases for which the formula is valid in a range for which both the expected value and variance are finite are when H is a cycle with attached trees and when H is a tree.

On the chromatic number of random d-regular graphs

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d < 2 ( k − 1 ) log ( k − 1 ) . From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k − 1 or k. If moreover d > ( 2 k − 3 ) log ( k − 1 ) , then the value k − 1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k + 1 . Our proof applies the small subgraph conditioning method to the number of equitable k-colourings, where a colouring is equitable if the number of vertices of each colour is equal.

Minimum degree conditions for large subgraphs

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turan's theorem [Turan, P., On an extremal problem in graph theory (in Hungarian), Matematiko Fizicki Lapok 48 (1941), 436–452]) or on finding spanning subgraphs (such as Dirac's theorem [Dirac, G.A., Some theorems on abstract graphs, Proc. London Math. Soc. s3-2 (1952), 69–81] or more recently work of Komlos, Sarkozy and Szemeredi [Komlos, J., G. N. Sarkozy and E. Szemeredi, On the square of a Hamiltonian cycle in dense graphs, Random Struct. Algorithms 9 (1996), 193-211; Komlos, J., G. N. Sarkozy and E. Szemeredi, Proof of the Seymour Conjecture for large graphs, Ann. Comb. 2 (1998), 43–60] towards a proof of the Posa-Seymour conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph. We contend that this neglect is unjustified. To support our contention we focus on the illustrative case of minimum degree conditions which guarantee squared-cycles of various lengths, but also offer results, conjectures and comments on other powers of paths and cycles, generalisations thereof, and hypergraph variants.

On the chromatic number of a random 5‐regular graph

AbstractIt was only recently shown by Shi and Wormald, using the differential equation method to analyze an appropriate algorithm, that a random 5‐regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5‐regular graph is asymptotically almost surely equal to 3, provided a certain four‐variable function has a unique maximum at a given point in a bounded domain. We also describe extensive numerical evidence that strongly suggests that the latter condition holds. The proof applies the small subgraph conditioning method to the number of locally rainbow balanced 3‐colorings, where a coloring is balanced if the number of vertices of each color is equal, and locally rainbow if every vertex is adjacent to at least one vertex of each of the other colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 157–191, 2009

Motif Analysis and the Periodic Structural Changes in an Organizational Email-Based Social Network

Network motifs are small subgraphs that reflect local network topology and were shown to be useful for creating profiles that reveal several properties of the network. In this work the motif analysis of the e-mail network of the Wroclaw University of Technology, consisting of over 4000 nodes was conducted. Temporal changes in the network structure during the period of 20 months were analysed and the correlations between global structural parameters of the network and motif distribution were found. These results are to be used in the development of methods dedicated for fast estimating of the properties of complex internet-based social networks.

Expanding network communities from representative examples

We present an approach to leverage a small subset of a coherent community within a social network into a much larger, more representative sample. Our problem becomes identifying a small conductance subgraph containing many (but not necessarily all) members of the given seed set. Starting with an initial seed set representing a sample of a community, we seek to discover as much of the full community as possible. We present a general method for network community expansion, demonstrating that our methods work well in expanding communities in real world networks starting from small given seed groups (20 to 400 members). Our approach is marked by incremental expansion from the seeds with retrospective analysis to determine the ultimate boundaries of our community. We demonstrate how to increase the robustness of the general approach through bootstrapping multiple random partitions of the input set into seed and evaluation groups. We go beyond statistical comparisons against gold standards to careful subjective evaluations of our expanded communities. This process explains the causes of most disagreement between our expanded communities and our gold-standards—arguing that our expansion methods provide more reliable communities than can be extracted from reference sources/gazetteers such as Wikipedia.

Separator-based data reduction for signed graph balancing

Polynomial-time data reduction is a classical approach to hard graph problems. Typically, particular small subgraphs are replaced by smaller gadgets. We generalize this approach to handle any small subgraph that has a small separator connecting it to the rest of the graph. The problem we study is the NP-hard Balanced Subgraph problem, which asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. The data reduction scheme unifies and generalizes a number of previously known data reductions, and can be applied to a large number of graph problems where a coloring or a subset of the vertices is sought. To solve the instances that remain after reduction, we use a fixed-parameter algorithm based on iterative compression with a very effective heuristic speedup. Our implementation can solve biological real-world instances exactly for which previously only approximations were known. In addition, we present experimental results for financial networks and random networks.

Moderate Deviations in a Random Graph and for the Spectrum of Bernoulli Random Matrices

We prove the moderate deviation principle for subgraph count statistics of Erdös-Rényi random graphs. This is equivalent in showing the moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gärtner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entries.

Avoiding small subgraphs in Achlioptas processes

AbstractFor a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r = 2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component.In this article, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles Ct, cliques Kt, and complete bipartite graphs Kt,t, in every Achlioptas process with parameter r ge 2. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

Exploration of Networks using overview+detail with Constraint-based cooperative layout

A standard approach to large network visualization is to provide an overview of the network and a detailed view of a small component of the graph centred around a focal node. The user explores the network by changing the focal node in the detailed view or by changing the level of detail of a node or cluster. For scalability, fast force-based layout algorithms are used for the overview and the detailed view. However, using the same layout algorithm in both views is problematic since layout for the detailed view has different requirements to that in the overview. Here we present a model in which constrained graph layout algorithms are used for layout in the detailed view. This means the detailed view has high-quality layout including sophisticated edge routing and is customisable by the user who can add placement constraints on the layout. Scalability is still ensured since the slower layout techniques are only applied to the small subgraph shown in the detailed view. The main technical innovations are techniques to ensure that the overview and detailed view remain synchronized, and modifying constrained graph layout algorithms to support smooth, stable layout. The key innovation supporting stability are new dynamic graph layout algorithms that preserve the topology or structure of the network when the user changes the focus node or the level of detail by in situ semantic zooming. We have built a prototype tool and demonstrate its use in two application domains, UML class diagrams and biological networks.

On the origin of distribution patterns of motifs in biological networks

BackgroundInventories of small subgraphs in biological networks have identified commonly-recurring patterns, called motifs. The inference that these motifs have been selected for function rests on the idea that their occurrences are significantly more frequent than random.ResultsOur analysis of several large biological networks suggests, in contrast, that the frequencies of appearance of common subgraphs are similar in natural and corresponding random networks.ConclusionIndeed, certain topological features of biological networks give rise naturally to the common appearance of the motifs. We therefore question whether frequencies of occurrences are reasonable evidence that the structures of motifs have been selected for their functional contribution to the operation of networks.

Insights into biological information processing: structural and dynamical analysis of a human protein signalling network

We present an investigation on the structural and dynamical properties of a ‘human protein signalling network’ (HPSN). This biological network is composed of nodes that correspond to proteins and directed edges that represent signal flows. In order to gain insight into the organization of cell information processing this network is analysed taking into account explicitly the edge directions. We explore the topological properties of the HPSN at the global and the local scale, further applying the generating function formalism to provide a suitable comparative model. The relationship between the node degrees and the distribution of signals through the network is characterized using degree correlation profiles. Finally, we analyse the dynamical properties of small sub-graphs showing high correlation between their occurrence and dynamic stability.

GraphCrunch: A tool for large network analyses

BackgroundThe recent explosion in biological and other real-world network data has created the need for improved tools for large network analyses. In addition to well established global network properties, several new mathematical techniques for analyzing local structural properties of large networks have been developed. Small over-represented subgraphs, called network motifs, have been introduced to identify simple building blocks of complex networks. Small induced subgraphs, called graphlets, have been used to develop "network signatures" that summarize network topologies. Based on these network signatures, two new highly sensitive measures of network local structural similarities were designed: the relative graphlet frequency distance (RGF-distance) and the graphlet degree distribution agreement (GDD-agreement).Finding adequate null-models for biological networks is important in many research domains. Network properties are used to assess the fit of network models to the data. Various network models have been proposed. To date, there does not exist a software tool that measures the above mentioned local network properties. Moreover, none of the existing tools compare real-world networks against a series of network models with respect to these local as well as a multitude of global network properties.ResultsThus, we introduce GraphCrunch, a software tool that finds well-fitting network models by comparing large real-world networks against random graph models according to various network structural similarity measures. It has unique capabilities of finding computationally expensive RGF-distance and GDD-agreement measures. In addition, it computes several standard global network measures and thus supports the largest variety of network measures thus far. Also, it is the first software tool that compares real-world networks against a series of network models and that has built-in parallel computing capabilities allowing for a user specified list of machines on which to perform compute intensive searches for local network properties. Furthermore, GraphCrunch is easily extendible to include additional network measures and models.ConclusionGraphCrunch is a software tool that implements the latest research on biological network models and properties: it compares real-world networks against a series of random graph models with respect to a multitude of local and global network properties. We present GraphCrunch as a comprehensive, parallelizable, and easily extendible software tool for analyzing and modeling large biological networks. The software is open-source and freely available at . It runs under Linux, MacOS, and Windows Cygwin. In addition, it has an easy to use on-line web user interface that is available from the above web page.

A Distributed SDP Approach for Large-Scale Noisy Anchor-Free Graph Realization with Applications to Molecular Conformation

We propose a distributed algorithm for solving Euclidean metric realization problems arising from large 3-D graphs, using only noisy distance information and without any prior knowledge of the positions of any of the vertices. In our distributed algorithm, the graph is first subdivided into smaller subgraphs using intelligent clustering methods. Then a semidefinite programming relaxation and gradient search method are used to localize each subgraph. Finally, a stitching algorithm is used to find affine maps between adjacent clusters, and the positions of all points in a global coordinate system are then derived. In particular, we apply our method to the problem of finding the 3-D molecular configurations of proteins based on a limited number of given pairwise distances between atoms. The protein molecules, all with known molecular configurations, are taken from the Protein Data Bank. Our algorithm is able to reconstruct reliably and efficiently the configurations of large protein molecules from a limited number of pairwise distances corrupted by noise, without incorporating domain knowledge such as the minimum separation distance constraints derived from van der Waals interactions.

Generalized quasirandom graphs

We prove that if a sequence of graphs has (asymptotically) the same distribution of small subgraphs as a generalized random graph modeled on a fixed weighted graph H, then these graphs have a structure that is asymptotically the same as the structure of H. Furthermore, it suffices to require this for a finite number of subgraphs, whose number and size is bounded by a function of | V ( H ) | .

On Small Contractible Subgraphs in 3-connected Graphs of Small Average Degree

We study the distribution of small contractible subgraphs in 3-connected graphs under local regularity conditions.

Probabilistic abstraction for model checking

The goal of model checking is to verify the correctness of a given program, on all its inputs. The main obstacle, in many cases, is the intractably large size of the program's transition system. Property testing is a randomized method to verify whether some fixed property holds on individual inputs, by looking at a small random part of that input. We join the strengths of both approaches by introducing a new notion of probabilistic abstraction, and by extending the framework of model checking to include the use of these abstractions. Our abstractions map transition systems associated with large graphs to small transition systems associated with small random subgraphs. This reduces the original transition system to a family of small, even constant-size, transition systems. We prove that with high probability, “sufficiently” incorrect programs will be rejected (ε-robustness). We also prove that under a certain condition (exactness), correct programs will never be rejected (soundness). Our work applies to programs for graph properties such as bipartiteness, k -colorability, or any ∃∀ first order graph properties. Our main contribution is to show how to apply the ideas of property testing to syntactic programs for such properties. We give a concrete example of an abstraction for a program for bipartiteness. Finally, we show that the relaxation of the test alone does not yield transition systems small enough to use the standard model checking method. More specifically, we prove, using methods from communication complexity, that the OBDD size remains exponential for approximate bipartiteness.